Optimal. Leaf size=12 \[ \frac{1}{2} \log \left ((x+2)^2+1\right ) \]
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Rubi [A] time = 0.0057401, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {372, 260} \[ \frac{1}{2} \log \left ((x+2)^2+1\right ) \]
Antiderivative was successfully verified.
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Rule 372
Rule 260
Rubi steps
\begin{align*} \int \frac{2+x}{1+(2+x)^2} \, dx &=\operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,2+x\right )\\ &=\frac{1}{2} \log \left (1+(2+x)^2\right )\\ \end{align*}
Mathematica [A] time = 0.0028449, size = 12, normalized size = 1. \[ \frac{1}{2} \log \left ((x+2)^2+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 12, normalized size = 1. \begin{align*}{\frac{\ln \left ({x}^{2}+4\,x+5 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05449, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, \log \left ({\left (x + 2\right )}^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42029, size = 32, normalized size = 2.67 \begin{align*} \frac{1}{2} \, \log \left (x^{2} + 4 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082141, size = 10, normalized size = 0.83 \begin{align*} \frac{\log{\left (x^{2} + 4 x + 5 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09248, size = 15, normalized size = 1.25 \begin{align*} \frac{1}{2} \, \log \left (x^{2} + 4 \, x + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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